Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #1141 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C22⋊D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#1141 |
| Fitting | C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ D8 |
| Sylow | C7 ⋊ C3×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22⋊D7 |
| commutator, cosocle | C4×C7 | C22×C3 |
| Frattini | C4 | C2×C3×D7 |
| Fitting | C3×C7×D4 | C2 |
| socle | C2×C3×C7 | D4 |
| hypercenter | C3×D4 | D7 |
| hypocenter | C7 | C3×D8 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,6,7,5,3)(9,12)(10,15)(13,14)(16,17,18);(2,3)(4,5)(6,7)(8,9)(10,12)(11,14)(13,15)66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1141, #10621042, #10621043, … |
| maximal normal subgroup of | D3×#1141, #10621042, #10621043, … |
| maximal quotient of | C3×#6381, C3×#6383, C3×#6400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 69);